Zero-free strips for the Riemann zeta-function derived from the Prime Number Theorem

We use the Prime Number Theorem to prove the existence of zero-free strips for the Riemann-zeta function. Precisely, we prove that there exists $\delta >0$ for which if $0\le r<\delta$ then $\zeta (s)\mathrm{\ne }0$ for Re$(s)>1-r$.